>Just want to suggest that the right question is not which leads most
>appropriately into real analysis or algebraic geometry or whatever. The
>right question about a foundational approach is: what sort of insight does
>it yield into the outstanding foundational problems of our time? Personally
>I believe these center about undecidability/independence results.
The problem is, how to you tell which are the "outstanding
foundational problems"? The argument will only recur at that level.
That said, undecidability/independence questions are surely are
among these problems.
>Which of the proposed foundational approaches is more likely to help in
>understanding/resolving this?
No choice between ZF and categorical set theory will help. Important
undecidability/independence questions are invariant under technical changes
in your set theory. For example, they do not depend on whether or not you
think "the intersection of the real numbers e and pi" is a well defined set.
The differences come in other places:
Categorical foundations suggest a wider range of topics. (Simpson and I
agree on this fact, but he feels it invalidates the claim to foundationality).
Categorical foundations are closer in style and methods to mainstream
mathematics. (Friedman considers mainstream mathematics intellectually corrupt.)
Categorical foundations emphasizes functions, and considers it meaningless
to talk about "membership" between arbitrary sets--e.g. it is meaningless to
ask whether the set of integers is a member of the set of symmetries of the
plane. (Friedman feels this violates the expectations of small children.)
Colin